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Indecomposability in field theory and applications to disordered systems and geometrical problems

Vasseur Romain

To cite this version:

Vasseur Romain. Indecomposability in field theory and applications to disordered systems and ge- ometrical problems. Mathematical Physics [math-ph]. Université Pierre et Marie Curie - Paris VI, 2013. English. �tel-00876155�

UNIVERSIT´E PIERRE ET MARIE CURIE - PARIS VI et

INSTITUT DE PHYSIQUE TH´EORIQUE - CEA/SACLAY Ecole Doctorale de Physique de la R´egion Parisienne - ED 107´

Th` ese de doctorat

Sp´ecialit´e : Physique Th´eorique

Indecomposability in field theory and applications to disordered systems and geometrical problems

pr´esent´ee par Romain Vasseur

pour obtenir le grade de Docteur de l’Universit´e Pierre et Marie Curie Th`ese pr´epar´ee sous la direction de Hubert Saleuret de Jesper Jacobsen

Soutenue le 27 Septembre 2013 devant le jury compos´e de :

Denis Bernard Examinateur Ecole Normale Sup´erieure´ John Cardy Rapporteur Oxford University

Matthias Gaberdiel Rapporteur ETH Z¨urich

Jesper Jacobsen Membre invit´e (co-directeur) Ecole Normale Sup´erieure´

Nicholas Read Examinateur Yale University

Hubert Saleur Directeur de th`ese CEA Saclay

Jean-Bernard Zuber Pr´esident du jury Universit´e Pierre et Marie Curie

Indecomposabilit´ e dans les th´ eories des champs et applications aux syst` emes d´ esordonn´ es et aux probl` emes g´ eom´ etriques

R´ esum´ e :

Les th´eories des champs conformes logarithmiques (LCFTs) sont cruciales pour d´ecrire le comportement critique de syst`emes physiques vari´es : les transitions de phase dans les syst`emes ´electroniques d´esordonn´es sans interaction (comme par exemple la transition entre plateaux dans l’effet Hall quantique entier), les points critiques d´esordonn´es dans les syst`emes statistiques classiques (comme le mod`ele d’Ising avec liens al´eatoires), ou encore les mod`eles g´eom´etriques critiques (comme la percolation ou les marches al´eatoires auto-´evitantes). Les LCFTs d´ecrivent des th´eories non unitaires, qui ne seraient probablement pas pertinentes dans le contexte de la physique des parti- cules, mais qui apparaissent naturellement en mati`ere condens´ee et en physique statis- tique. Sans cette condition d’unitarit´e, toute la puissance alg´ebrique qui a fait le succ`es des th´eories conformes est fortement compromise `a cause de “l’ind´ecomposabilit´e” de la th´eorie des repr´esentations sous-jacente. Ceci a pour cons´equence de modifier les fonc- tions de corr´elation alg´ebriques par des corrections logarithmiques, et r´eduit s´ev`erement l’espoir d’une classification g´en´erale.

Le but de cette th`ese est d’analyser ces th´eories logarithmiques en ´etudiant leur r´egularisation sur r´eseau, l’id´ee principale ´etant que la plupart des difficult´es alg´ebriques caus´ees par l’ind´ecomposabilit´e sont d´ej`a pr´esentes dans des syst`emes de taille finie.

Notre approche consiste `a consid´erer des mod`eles statistiques critiques avec matrice de transfert non diagonalisable (ou des chaˆınes de spins critiques avec Hamiltonien non diagonalisable) et d’analyser leur limite thermodynamique `a l’aide de diff´erentes m´ethodes num´eriques, alg´ebriques et analytiques. On explique en particulier comment mesurer num´eriquement les param`etres universels qui caract´erisent les repr´esentations ind´ecomposables qui apparaissent `a la limite continue. L’analyse d´etaill´ee d’une vaste classe de mod`eles sur r´eseau nous permet ´egalement de conjecturer une classification de toutes les LCFTs chirales pertinentes physiquement, pour lesquelles la seule sym´etrie est donn´ee par l’alg`ebre de Virasoro. Cette approche est aussi partiellement ´etendue aux th´eories non chirales, avec une attention particuli`ere port´ee au probl`eme bien connu de la formulation d’une th´eorie des champs coh´erente qui d´ecrirait la percolation en deux dimensions. On montre que les mod`eles sur r´eseaux p´eriodiques ou avec bords peuvent ˆetre reli´es alg´ebriquement seulement dans le cas des mod`eles minimaux, impliquant des cons´equences int´eressantes pour les th´eories des champs sous-jacentes. Un certain nombre d’applications aux syst`emes d´esordonn´es et aux mod`eles g´eom´etriques sont

´egalement abord´ees, avec en particulier une discussion d´etaill´ee des observables avec comportement logarithmique au point critique dans le mod`ele de Potts en dimension arbitraire.

Mots clefs :Th´eories des Champs Conformes Logarithmiques, Repr´esentations inde- composables, Mod`eles sur r´eseau, Syst`emes d´esordonn´es, Probl`emes g´eom´etriques.

Indecomposability in field theory and applications to disordered systems and geometrical problems

Abstract:

Logarithmic Conformal Field Theories (LCFTs) are crucial for describing the crit- ical behavior of a variety of physical systems. These include phase transitions in dis- ordered non-interacting electronic systems (such as the transition between plateaus in the integer quantum Hall effect), disordered critical points in classical statistical mod- els (such as the random bond Ising model), or critical geometrical models (such as polymers and percolation). LCFTs appear when one has to give up the unitarity con- dition, which is natural in particle physics applications, but not in statistical mechanics and condensed matter physics. Without unitarity, the powerful algebraic approach of conformal invariance encounters formidable technical difficulties due to ‘indecompos- ability’. This in turn yields logarithmic corrections to the power-law correlations at the critical point, and prevents the use of general classification techniques that have proven so powerful in the unitary case.

The goal of this thesis is to understand LCFTs by studying their lattice regular- izations, the crucial point being that most algebraic complications due to indecompos- ability occur in finite size systems as well. Our approach is thus to consider critical statistical models with non-diagonalizable transfer matrices, or gapless quantum spin chains with non-diagonalizable hamiltonians, and to study their scaling limit by uti- lizing a variety of algebraic, numerical and integrable techniques. We show how to measure numerically universal parameters that characterize the indecomposable rep- resentations of the Virasoro algebra which emerge in the thermodynamic limit. An extensive understanding of a wide class of lattice models allows us to conjecture a ten- tative classification of all possible (chiral) LCFTs with Virasoro symmetry only. This approach is partially extended to the bulk case, for which we discuss how the long- standing bulk CFT formulation of percolation can be tackled along these lines. We also argue that boundary and periodic lattice models can be related algebraically only in the case of minimal models, and we work out the consequences for the underlying boundary and bulk field theories. Several concrete applications to disordered systems and geometrical problems are discussed, and we uncover a large class of geometrical observables in the Potts model that behave logarithmically at the critical point.

Key words: Logarithmic Conformal Field Theory, Indecomposable representations, Temperley-Lieb algebra, Lattice models, Disordered systems, Geometrical problems.

Remerciements

Je voudrais commencer par remercier John Cardy et Matthias Gaberdiel, qui ont accept´e la p´enible tache de rapporteur, ainsi que les examinateurs, Denis Bernard, Nicholas Read et Jean-Bernard Zuber. Merci pour votre aide et vos conseils pour am´eliorer mon manuscrit. Je suis tr`es conscient du travail qu’ˆetre membre d’un comit´e de th`ese repr´esente, et je vous suis extrˆemement reconnaissant d’avoir accept´e de braver des heures de transport pour venir assister `a ma soutenance.

Je remercie aussi Hubert et Jesper, qui en plus de leur qualit´es scientifiques indis- cutables, m’ont beaucoup appris et aid´e dans bien des domaines, en particulier dans mes moments de doute, ou pendant la p´eriode difficile des recherches de postdocs.

Merci ´egalement de m’avoir soutenu durant mon changement de th´ematique apr`es ma premi`ere moiti´e de th`ese, mˆeme si cela repr´esentait un certain nombre de risques et de difficult´es pour vous aussi. Je sais que j’ai pu rˆaler `a certains moments, mais je suis vraiment ravi d’avoir fait ma th`ese avec vous deux, vous avez toujours ´et´e disponibles et `a l’´ecoute en cas de probl`eme : travailler avec vous durant ces trois ann´ees a toujours

´et´e un plaisir. Je tiens ´egalement `a remercier tout particuli`erement Azat Gainutdinov qui a ´et´e postdoc `a Saclay durant toute ma th`ese : j’ai beaucoup appr´eci´e travailler avec toi sur tous ces projets, ¸ca a ´et´e tr`es stimulant et b´en´efique pour moi.

J’ai eu la chance d’avoir des bureaux dans deux laboratoires, au LPTENS et `a l’IPhT au CEA, ce qui me semble ˆetre un luxe `a l’heure ou l’espace disponible pour les th´esards est chaque ann´ee plus restreint. Merci `a tous les gens qui font la vie de ces laboratoires, je pense notamment `a tous les membres du staff administratif, mais aussi au personnel informatique et `a tous les chercheurs avec qui j’ai eu l’opportunit´e de discuter. Merci `a Olivier Golinelli et `a St´ephane Nonnenmacher pour leurs conseils. Je tiens ´egalement `a remercier tous les th´esards et les postdocs qui ont crois´e mon chemin

`a Saclay (dans le d´esordre Piotr – sans toi Alexandre m’aurait encore plus empˆech´e de travailler – et Julien – sans toi, j’aurais sans doute oubli´e des noms dans cette liste – les anciens du M2, et aussi Thiago, Jean-Marie, Benoit, R´emi, Bruno, Richard, Guillaume, H´el`ene, Alexander, Hanna, Antoine, J´erˆome et tous les th´esards/postdocs qui continuent de manger ensemble le midi, ainsi que Katya, Nicolas et Thomas qui ont partag´e mon bureau), `a l’ENS ou `a Jussieu (Michele, Tristan, Fabien, S´ebastien, Alexander, Julius et `a tous ceux qui ont particip´e de pr`es ou de loin au s´eminaire jeunes cond-mat, qui je l’esp`ere continuera pendant de nombreuses ann´ees), ou au hasard

d’une conf´erence (je pense surtout `a Jacopo et Loic r´ecemment). Il m’est impossible de citer tous les noms ici, mais je souhaite remercier tout particuli`erement Alexandre bien sˆur, pour toutes nos discussions scientifiques ou non, et Swann pour les sandwiches Libanais du mercredi et sa compagnie durant le M2 (je me rappellerai toujours du

“that’s my spot” avec Pilou). Scu : malgr´e les nombreuses heures pendant lesquelles tu m’as empˆech´e de travailler, je reste persuad´e qu’un certain nombre de r´esultats de cette th`ese auraient sans doute mis plus de temps `a prendre forme sans nos innombrables discussions ! Enfin, je souhaite aussi remercier les th´esards anciens ou actuels de Jesper et Hubert : Eric (merci pour tes commentaires sur ce manuscrit !) et Roberto qui ont

´et´e th´esards en mˆeme temps que moi, mais aussi J´erˆome, Constantin et Yacine pour les discussions scientifiques ou non que nous avons pu avoir.

Il va sans dire que C´edric, Wahb et Alexandre tiennent une place tr`es particuli`ere dans ces remerciements : sans toutes nos soir´ees restau/pizzas/fondues/sushis/coktails depuis l’ENS Lyon, et sans nos vacances `a New York et dans l’ouest am´ericain pour notre mariage avec Mandy, ces ann´ees de th`ese auraient sans doute ´et´e moins sympa- thiques (mais probablement plus b´en´efiques pour ma ligne). Je suis tr`es content que nous ayons suivi des parcours similaires durant toutes ces ann´ees, votre pr´esence va sans aucun doute me manquer l’ann´ee prochaine.

Je remercie ma m`ere et Philippe pour leur soutien sans faille durant ma th`ese, mais aussi avant en pr´epa et `a l’ENS, je ne sais pas ce qu’on aurait fait sans vous. Merci pour tous ces moments r´econfortants autour d’une bonne pizza ! Merci aussi `a Anne, Dom, Francine et Joel pour avoir fait le d´eplacement pour ma soutenance, je suis ravi que vous soyez l`a. Je profite aussi de ces quelques lignes pour penser `a Sara et Raphael qui n’ont pas beaucoup vu leur fr`ere durant l’´ecriture de cette th`ese, et aussi `a Laure.

Enfin, mes derniers remerciements vont tout naturellement `a ma femme Mandy, qui a v´ecu ses ann´ees de th`eses avec moi et qui me suit depuis toutes ces ann´ees d’´etudes.

Inutile d’en ´ecrire des pages, je n’en serais pas l`a sans toi, et j’ai hˆate de profiter de notre nouvelle vie en Californie avec Newton !

Note to the reader

This manuscript contains a review of my doctoral work on Logarithmic Conformal Field Theory that corresponds more or less to the first two years of my PhD thesis.

More recently, I have been involved in various projects related to quantum quenches and entanglement in quantum impurity problems, which would have arguably deserved one or two chapters in this thesis manuscript. However, for the sake of consistency and brevity, I have decided not to include here this part of my work. I refer the interested reader to the original papers on that topic that are included at the end of this thesis.

Before embarking on our journey through the various aspects of Logarithmic Con- formal Field Theories, I would like to warn the reader that I have chosen to address this admittedly very technical topic in a rather loose way, without paying too much attention to precise mathematical definitions. The words ‘modules’ and ‘representa- tions’ are used interchangeably, very technical mathematical concepts such as ‘tiltings’

or ‘projectives’ are defined and used in a ‘physical way’ etc. I hope that the reader fond of mathematical rigor will forgive me for that pedagogical choice.

Contents

1 Introduction: logarithmic correlations in condensed

matter physics 1

1.1 Non-unitarity and LCFTs in statistical mechanics and condensed matter. . . 3

1.2 A little bit of history . . . 6

1.3 Why lattice models? . . . 7

1.4 Organization of the manuscript . . . 9

2 A short introduction to Logarithmic Conformal Field Theory 11

2.1.1 Conformal invariance in 2D, primary operators and Ward identity . . 12

2.1.2 Logarithmic operators and two-point functions . . . 15

2.1.3 Indecomposability parameters . . . 18

2.2 𝑐→0 catastrophe, differential equations and generalizations . . . 20

2.2.1 𝑐→0 catastrophe . . . 20

2.2.2 Generalization and formulas for indecomposability parameters . . . 23

2.2.3 Boundary dilute polymers four-point functions . . . 26

2.3 Intermezzo: Indecomposable representations and 𝐺𝐿(1∣1) . . . 28

2.3.1 Defining relations and irreducible representations . . . 28

2.3.2 A first look at indecomposability . . . 29

2.3.3 Tensor products . . . 30

2.4 Indecomposability in field theory: the example of symplectic fermions . . . 32

2.4.1 Symplectic fermions . . . 33

2.4.2 Hilbert space and zero-dimensional logarithmic operators. . . 34

2.4.3 A first step towards Virasoro staggered modules . . . 37

2.5 Indecomposable Virasoro representations . . . 40

2.5.1 Virasoro algebra and Verma modules . . . 40

2.5.2 Verma modules theory . . . 41

2.5.3 Virasoro staggered modules and indecomposability parameters . . . 43

2.5.4 A word on fusion . . . 48

3 From the Temperley-Lieb algebra to Virasoro:

indecomposability in lattice models 49

3.1 Temperley-Lieb algebra, XXZ spin chain, loop models and supersymmetry . . . 49

3.1.1 Temperley-Lieb algebra, transfer matrix and Hamiltonian . . . 50

3.1.2 XXZ spin chain . . . 51

3.1.3 Potts and dense loop models . . . 52

3.1.4 Supersymmetric models and sigma models . . . 55

3.1.5 A remark on dilute models and other representations . . . 59

3.2 TL representation theory and Hilbert space decomposition . . . 60

3.2.1 Reduced states and standard modules . . . 60

3.2.2 Generic case: decomposition of the partition function . . . 61

3.2.3 Generic case: algebraic analysis of the spectrum, commutant and bimodules . . . 63

3.2.4 Representation theory of the Temperley-Lieb algebra . . . 65

3.2.5 Hilbert space decomposition in the root of unity case . . . 67

3.3 Scaling limit and Virasoro representations . . . 69

3.3.1 Finite size scaling . . . 69

3.3.2 Critical exponents, Kac modules and lattice Virasoro modes . . . . 70

3.3.3 Hilbert space structure and bimodules in the limit. . . 72

3.3.4 Virasoro staggered modules from the lattice . . . 74

3.3.5 Some consequences of the bimodule structure: differential equations . 75 3.4 Lattice indecomposability parameters . . . 76

3.5 Lattice fusion rules. . . 79

3.5.1 Fusion on the lattice and in the continuum . . . 79

3.5.2 𝑐→0 catastrophe on the lattice . . . 81

3.5.3 Systematic calculations and general results . . . 82

4 Blob algebra, braid translator and bulk LCFTs 85

4.1.1 Blob algebra and Verma modules . . . 86

4.1.2 Representation theory: from the Blob algebra to Virasoro . . . 91

4.1.3 Towards a classification of Virasoro indecomposable representations . 93 4.2 Periodic lattice models and bulk LCFTs . . . 94

4.2.1 Periodic version of the TL algebra, standard modules and operator content . . . 95

4.2.2 Periodic 𝔤𝔩(1∣1) spin chain and lattice indecomposability parameters . 98 4.2.3 Measure of𝑏 in periodic percolation . . . 102

4.2.4 Towards a bulk𝑐= 0 LCFT for percolation . . . 110

4.3 From boundary to bulk LCFTs: braid translator . . . 112

4.3.1 Braid translator . . . 112

4.3.2 Braid translation of minimal models. . . 114

4.3.3 Braid translation of Logarithmic CFTs . . . 117

5 Logarithmic structure of geometrical models and

disordered systems in dimension 𝑑 ≥ 2 121

5.1 Critical disordered systems: supersymmetry, replicas and 𝑐= 0 LCFTs . . . 122

5.1.1 Disordered systems and replicas . . . 122

5.1.2 Disordered systems and SUSY . . . 125

5.2 Operator content of the Potts model in 𝑑 dimensions. . . 126

5.2.1 𝑆𝑄 representation theory . . . 126

5.2.2 Watermelon operators𝑡^(𝑘,𝑁)𝑎1,...,𝑎𝑘 . . . 128

5.2.3 Correlation functions: discrete results . . . 129

5.3 Logarithmic correlations: spanning trees and forests, percolation and sub- leading operators . . . 133

5.3.1 𝑄→0: spanning trees and spanning forests . . . 133

5.3.2 Percolation (𝑄→1 limit) . . . 139

5.3.3 An example of logarithm in the Ising model (𝑄→2 limit) . . . 142

5.4 A short overview of the generalization to the 𝒪(𝑛) model . . . 143

A The Lie superalgebras 𝔤𝔩(1 ∣ 1) and 𝔰𝔩(2 ∣ 1) and some of their representations 151

A.1.1 Defining relations. . . 152

A.1.2 Fundamental and Dual representations in Fock space . . . 152

A.1.3 Finite dimensional representations of𝔤𝔩(1∣1) . . . 153

A.2 The Lie superalgebra 𝔰𝔩(2∣1) and its representations. . . 153

A.2.1 Defining relations. . . 153

A.2.2 Fundamental and Dual representations in Fock space . . . 154

A.2.3 Finite dimensional representations . . . 154

Bibliography 157

Chapter 1

Introduction: logarithmic correlations in condensed matter physics

This thesis manuscript aims at providing a rather self-contained introduction to the field of Logarithmic Conformal Field Theory (LCFT), with of course a particular emphasis on the work of the author. The techniques of Conformal Field Theory (CFT) have proven very efficient over the past decades at describing critical phenomena. In a typical critical physical system, the correlations within the sample do not decay expo- nentially with a characteristic correlation length𝜉, but rather algebraically with a set of universal critical exponents that may be the same for very different physical problems, depending on the symmetries and the dimensionality of the system rather than on mi- croscopic details. In the well-known example of the ferromagnetic/antiferromagnetic transition, described in its simplest version by the Ising model (see Fig. 1.1), the corre- lation length diverges as𝜉 ∼ ∣𝑇−𝑇𝑐∣^−𝜈 when the temperature approaches the so-called Curie temperature 𝑇𝑐, with a critical exponent 𝜈 (𝜈 = 1 in 𝑑 = 2 dimensions, and 𝜈 = ¹₂ in mean-field theory). At the critical point 𝑇 = 𝑇𝑐, the two-point function of the spin operator 𝑆(𝑟) scales as

⟨𝑆(⃗𝑟₁)𝑆(⃗𝑟₂)⟩ ∼ ∣⃗𝑟₁−⃗𝑟₂∣^{2−𝑑−𝜂}, (1.1) where 𝜂 is yet another critical exponent, with 𝜂 = ¹₄ in two dimensions, as computed by Wu and Chen [1, 2], more than twenty years after the exact calculation of the free energy by Onsager [3]. The power-law behavior of eq. (1.1) can actually be traced back to the expected scale invariance at the critical point. As it turns out, the symmetry of the critical point is much larger and contains all transformations preserving local scale invariance; these are called conformal transformations, and they include for example rotation and translation symmetries. Quantum Field Theories with such conformal symmetry describe Renormalization Group fixed points, and are called Conformal Field Theories. CFTs are very constrained by symmetry, especially in two dimensions (2D) where the Lie algebra of conformal transformations becomes infinite, giving rise to the

𝑇

𝑇 = 0 𝑇 = ∞

b b b

Figure 1.1: Phase diagram of the two-dimensional Ising model. The system flows, in the sense of the renormalization group, to a massive disordered phase for 𝑇 > 𝑇𝑐. At the critical point 𝑇 =𝑇𝑐, the low energy behavior of the Ising model is described by a Conformal Field Theory with central charge 𝑐= ¹₂.

celebrated Virasoro algebra

[𝐿𝑛, 𝐿𝑚] = (𝑛−𝑚)𝐿𝑛+𝑚+ 𝑐

12𝑛(𝑛²−1)𝛿𝑛+𝑚,0, (1.2)

where𝑐∈ℝ(or𝑐∈ℂa priori) is the central charge. Since the seminal paper [4], many exact results for critical systems have been obtained using conformal invariance tech- niques. Among the most famous examples are the unitary minimal modelsℳ(𝑝, 𝑝+ 1), with central charge [5, 6]

𝑐= 1− 6

𝑝(𝑝+ 1), with 𝑝≥ 3 integer, (1.3) obtained from the classification of irreducible representations of the Virasoro alge- bra (1.2). In that series, 𝑝= 3 corresponds to the Ising model, 𝑝= 4 to the tricritical Ising model, 𝑝= 5 to the 3-state Potts model, etc. It is also worth pointing out that as 𝑝 increases, more and more relevant operators are allowed and the corresponding critical points are obtained through fine tuning of more and more parameters, so that only small values of 𝑝 are actually interesting for practical applications. CFTs were also found to have a very broad range of applications in condensed matter and quan- tum impurity physics (related to Boundary CFTs [7]), from the Kondo effect [8], the Fermi Edge singularity [9] or the wave functions describing Fractional Quantum Hall states [10] to, more recently, entanglement entropy [11], quantum quenches [12, 13] or non-equilibrium heat current [14] calculations in 1D quantum spin chains.

Despite its success, many interesting physical applications of CFT actually involve much more complicated field theories whose understanding is not complete. These include for instance quantum critical points in disordered systems of non-interacting fermions – an example being the long sought-after theory of the transition between

plateaus in the Integer Quantum Hall Effect (IQHE) [15] (seee.g.[16] for a review from a CFT perspective and references therein), two-dimensional geometrical problems such as self-avoiding walks and percolation [17], or critical systems with quenched disorder in general [18, 19]. Those arguably interesting physical problems turn out to be described by daunting CFTs that show unusual features such as non-unitarity, which means one has to deal with non-Hermitian Hamiltonians and negative norm-square states, or non- rationality, which implies that the scaling operators in the theories cannot be simply described by a finite set of primary operators – all remaining operators being roughly derivatives (descendants) of these few primary fields. As we shall discuss shortly, non- unitarity opens the door to indecomposability, a crucial feature that ultimately lead to logarithmic corrections to algebraic correlations such as (1.1)

⟨𝑆(⃗𝑟˜ 1) ˜𝑆(⃗𝑟2)⟩ ∼ ∣⃗𝑟1−⃗𝑟2∣^−2Δ(𝛼+𝛽log∣⃗𝑟1−⃗𝑟2∣), (1.4) where Δ is the scaling dimension of the field ˜𝑆. It is important to emphasize the fact the logarithmic corrections in (1.4) appearatthe critical point; in particular, those are not sub-leading corrections produced by marginally irrelevant operators as in the 𝜙⁴ field theory in 𝑑 = 4 dimensions [20] or in the XY model at the Kosterlitz-Thouless point [21]. The existence of logarithmic fields like ˜𝑆(⃗𝑟) requires the generalization of the usual scale invariance principle of a Quantum Field Theory, which translates into complicatedindecomposablerepresentations from the point of view of the Virasoro algebra. This shall be explained in more detail in the following.

The remainder of this introduction will be devoted to some examples of physical systems expected to have logarithmic correlations. We will also give some motivations for the lattice methods that shall be used extensively throughout this thesis.

1.1 Non-unitarity and LCFTs in statistical mechan- ics and condensed matter

Whereas non-unitarity can probably be considered as non-physical in the context of particle physics and traditional Quantum Mechanics, it turns out to be a quite natural feature in statistical mechanics, geometrical problems and condensed matter physics.

For instance, the description of statistical properties of geometrical problems such as percolation (see Fig. 1.2) or self-avoiding walks (SAWs, also known as dilute polymers) involves features that are more complicated than those of the minimal models (1.3).

One might reasonably think that this is in fact due to the non-local nature of these geometrical problems. Indeed, the main difference between, say the Ising model and percolation, is the intrinsic non-locality of the physical observables in percolation, where one is interested in connectivity probabilities of percolation clusters rather than in local spin or energy observables. However, this point of view is somehow misleading, as genuine non-locality would obviously spoil the Field Theory description of the problem.

In fact, those geometrical problems are only superficially non-local, as the non-locality can be traded for non-unitarity and complex Boltzmann weights [22], by reformulating them in terms of vertex models. Giving a weight𝑛 = 2 cos𝛾to a loop on the honeycomb

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

b b b b b b b b b b b b

Figure 1.2: Two-dimensional percolation configuration on the square lattice. Bonds are occupied with probability𝑝, and at the critical point𝑝=𝑝𝑐 = ¹₂, the scaling properties of percolation clusters are described by a 𝑐= 0 (L)CFT.

lattice can for example be done by orienting the loops, and giving an elementary complex weight e^±𝑖𝛾/6 to each left/right turn. Summing over both loop orientations thus gives each loop a fugacity 2 cos (6×𝛾/6) = 𝑛 as requested, since the number of left and right turns in a closed loop can only differ by 6. One therefore recovers locality at the price of giving up the natural probabilistic interpretation of the models.

From the 2D CFT perspective, the scaling properties of percolation and SAWs can be described by correlation functions and critical exponents given by a CFT with central charge 𝑐= 0. As we shall see later on, such 𝑐= 0 must be non-unitary, and actually logarithmic, in order to be non-trivial. The only unitary CFT with 𝑐 = 0 indeed has a unique observable, the identity operator with scaling dimension Δ = 0 [6]. It is also worth mentioning that even the Ising model can be considered as a LCFT, provided that one includes (apparently) non-local observables in the theory, such as fields measuring the probability that two spins belong to the same spin cluster for example. In that sense, logarithmic CFTs can actually be thought of as extensions of the minimal models outside the minimal Kac table, thus including more operators in the theory.

A few important remarks should be emphasized at this point.

∙ In the case of percolation or SAWs, the logarithmic nature of the LCFTs did not deter physicists from using conformal invariance to compute interesting physi- cal quantities such as critical exponents [23, 24] and crossing probabilities [25].

However, the full CFT description of such geometrical problems remains sadly unknown [26], as are almost all bulk correlation functions.

∙ Although we have insisted on the fact that one of the main aspects of a LCFT is non-unitarity, there exist some very simple non-unitary CFTs which can be tackled quite easily. Minimal models can indeed be extended to non-unitary theories, a well-known example being the Yang-Lee singularity CFTℳ(2,5) with

Figure 1.3: Plateaus for the Hall resistance and peaks of the Ohmic resistance in the integer quantum Hall effect. Neighboring values of 𝑖 are separated by a quantum critical point, whose properties are described by a𝑐= 0 2D (L)CFT. (figure taken from http://www.nobelprize.org/nobel prizes/physics/laureates/1998/press.html).

central charge𝑐=−²²₅ , describing the Ising model above its critical temperature in a non-zero, purely imaginary magnetic field [27]. Therefore, although a LCFT mustbe non-unitary, non-unitary CFTs do not have to be logarithmic and can be described thanks to a finite number of irreducible Virasoro representations just like unitary minimal models.

∙ The key feature of a LCFT is rather indecomposability, which means, from the point of view of Virasoro representation theory, that one has to deal with complicated reducible representations that cannot be decomposed into a direct sum of irreducible representations. In physical terms, this will imply the non- diagonalizability of the scale transformation generator – the Hamiltonian in a 2D CFT. This is allowed because the scale transformation generator does not have to be Hermitian in a non-unitary theory. We will come back to this in the next chapter.

Disordered systems provide another class of physical systems whose critical points are expected to be described by LCFTs. Of course, impurities and random disorder break conformal invariance in general, but upon averaging over disorder configurations, it is reasonable to expect that conformal invariance could be restored for specific values of the disorder strength and other physical parameters such as the temperature. We will call disordered or random critical points the resulting conformally invariant RG fixed points. The main issue when facing a problem with quenched disorder is to

average correlation functions such as

⟨𝒪⟩= 1

𝑍[{ℎ(⃗𝑟)}]Tr𝜙

(𝒪e^{−𝐻[𝜙,{ℎ(⃗𝑟)}]})

, (1.5)

over some quenched disordered variable ℎ(⃗𝑟). One solution to get rid of the partition function in the numerator of (1.5) is to consider 𝑛∈ℤ copies (replicas) of the system, and then to take a formal 𝑛→0 limit to recover physical results. Another alternative in the case of non-interacting system is to find other degrees of freedom 𝜓 such that 𝑍⁻¹ = Tr𝜓e^{−𝐻[𝜓]}. This is the essence of the supersymmetry (SUSY) approach to disordered systems [28], where in practical applications, 𝜙 and 𝜓 are bosonic and fermionic degrees of freedom, respectively. In both approaches, one ends up computing averaged observables using an effective field theory with trivial partition function, thus implying [29] the vanishing of the central charge 𝑐= 0 for disordered fixed points [18, 19]. Among the examples of problems described by 𝑐 = 0 LCFT is the transition between plateaus in the IQHE mentioned above [16] (see Fig. 1.3), or for instance, the so-called Nishimori point in the two-dimensional random-bond Ising model (see e.g. [30]). Other examples include the Spin Quantum Hall Transition – related to classical percolation [31], or Dirac fermions in a random SU(𝑁) gauge potential [32–

34].

Of course, physical applications of LCFTs are not restricted to condensed matter physics. For example, let us also mention Abelian sandpile models [35, 36], 2D tur- bulence [37, 38], or the AdS/CFT correspondence where LCFTs describe the massless limit of non-linear sigma models with non-compact target spaces [39, 40]. LCFTs also appear as duals of ‘logarithmic gravity’ theories in the AdS/LCFT correspon- dence [41, 42], and they also play an important role in 4D gauge theories [43].

Given the broad range of applications of LCFTs, it seems quite natural to try to push further our understanding of such quantum field theories, in the hope that someday we will be able to classify and handle them as well as we do minimal CFTs.

In this thesis, we will mostly focus on lattice models and point out how they can help in getting LCFTs under control. However, before we turn to this lattice regularization approach(orlattice approach), let us make a short historical detour to understand how and why physicists got interested in LCFTs in the first place.

1.2 A little bit of history

It is of course not the purpose of this short paragraph to provide a detailed ac- count of all the contributions to the LCFT field, but rather to point out that the key ideas of Logarithmic CFTs came out of rather different communities, ranging from pure mathematics and string theory to condensed matter. The first observation of logarithmic terms and indecomposability in CFTs probably goes back to the work of Rozansky and Saleur [44] in 1992, who studied the Wess-Zumino-Witten (WZW) model on the supergroup GL(1∣1). This was followed shortly after by the pioneering paper of Gurarie [45] in 1993, who first introduced the concept of logarithmic operators and explained how these were compatible with conformal invariance. Although it would be

fair to say that LCFTs remained mostly unknown to most theoretical physicists at the time, several key papers contributed to the birth of the LCFT field a few years later, in 1996. First, logarithmic operators were shown by Caux, Kogan and Tsvelik to appear in the problem of Dirac Fermions in a random gauge potential [32]. At the same time, Gaberdiel and Kaush [46, 47], and Flohr [48], studied indecomposable fusion rules and logarithmic operators with motivations rather far from condensed matter physics and disordered systems. The same year again, the mathematician Rohsiepe uploaded on arXiv his preliminary work on Virasoro staggered modules [49], which were realized to be of crucial importance to LCFTs many years later. A few years afterwards, Gurarie introduced his𝑏-number for𝑐= 0 CFTs [50], which turned out to be closely related to the apparently very different indecomposability parameters studied a bit earlier in [46].

At the time, the number𝑏was thought to be a sort of new “central charge”, that would allow one to distinguish between different 𝑐 = 0 CFTs. Right about the same time, Kausch introduced the theory of symplectic fermions [51, 52], which remains even now one of the few exactly solvable logarithmic theories, for which everything is under con- trol. Cardy then showed that logarithmic corrections in disordered systems and in polymers could be understood in terms of limits within a replica approach [18, 53], thus providing a simple physical mechanism for the appearance of logarithmic correla- tions in disordered systems in general. Disordered systems and 𝑐= 0 CFTs were also studied by Gurarie and Ludwig [19, 54], who computed for the first time the allowed values for 𝑏 in the boundary versions of polymers and percolation.

It is now well-accepted that LCFTs are not characterized by a single parameter 𝑏, but rather by a complex structure of indecomposable Virasoro representations, with an infinite set of indecomposability parameters akin to 𝑏 characterizing their struc- ture. Over the past few years, two lines of thought have been considered. The first one is to deal directly with complicated indecomposable Virasoro representations (see e.g. [55–60]). The second approach is somewhat more concrete, and consists in study- ing thoroughly lattice models whose continuum limit is described by LCFTs. This is this ‘lattice approach’ that we shall analyze in details in the following.

1.3 Why lattice models?

Most of our understanding of ordinary (non-logarithmic, rational) CFTs came from the classification of irreducible representations of the Virasoro algebra. The null-vector conditions in Virasoro representation theory strongly constrain the operator content of minimal CFTs, and they also allow one to compute correlation functions through differential equationsetc[4–6, 61]. LCFTs, on the other hand, though still constrained by representation theory – at least compared to completely irrationnal CFTs with generic central charge, remain very poorly understood mainly because the involved (indecomposable) modules of the Virasoro algebra have a very complicated structure.

It is actually believed that it is impossible to classify such indecomposable modules;

this is why the representation theory of Virasoro is said to be wild[62], which roughly means that it is as complicated as it can be. It might therefore be hopeless to try to solve LCFTs by classifying Virasoro representations, although several partial results

have emerged in that direction recently¹.

Although it might appear almost as a bit of a heresy in the CFT world, LCFTs seem so complicated that another possibility is to turn to specific examples, from which one might hope to extract generic features. WZW models with super target spaces do provide ‘simple’ examples of LCFTs [63–65], but although some interesting lessons can be learned from them, all the known examples seem to be very closely related [66], and their features far from being generic.

Another approach that has proven very helpful consists in considering lattice models as lattice regularizations of LCFTs, that is, lattice models whose continuum limit is described by LCFTs. This may seem completely hopeless at first sight, as lattice models are in principle much more complicated than field theories. In particular, conformal invariance obviously holds only in the continuum limit. However, it was realized over the years that many features of the continuum limit already appear on small lattice systems, albeit in some finite dimensional forms. This was first observed quite a while ago using quantum groups by Pasquier and Saleur [67], although it was formalized as an efficient tool to study LCFTs only a few years ago by Read and Saleur [68, 69], and independently with less algebraic emphasis, by Pearce, Rasmussen and Zuber [70].

The point is that most of the algebraic features that make LCFTs so complicated, such as indecomposability, are already present in finite lattice systems. In general, we would like to get a handle on LCFTs that describe fixed points of interacting, non- unitary, field theories with well defined local actions. If such LCFTs do exist, it is reasonable to expect that they can be realized as continuum limit (or scaling limit) of lattice models, such as quantum spin chains with local interactions. The Hamiltonian densities form a lattice algebra whose representation theory is well under control, and as the continuum limit is taken, one expects conformal symmetry to emerge, and this lattice algebra to tend to Virasoro in some sense that remains to be made more precise.

As we shall argue, Virasoro indecomposable modules and their fusion rules can be seen as scaling limits of lattice representations, and much crucial physical information, including indecomposability parameters or 𝑏-numbers generalizing Gurarie’s [50], can be recovered from the lattice. The lattice approach that we shall describe in this thesis mostly relies on the original work of Read and Saleur [68, 69], but we will try to connect our results to other approaches whenever possible.

Yet another good reason to study lattice models in the LCFTs context is that after all, at least for a condensed matter/statistical physicist, LCFTs can be considered as physical only if they do describe the low energy limit of some microscopic model of physical relevance. It might of course be important to construct 𝑐= 0 theories from a purely abstract point of view, but arguably, the ultimate goal is to get under control the CFTs describing physical systems such as percolation or disordered systems. It is therefore quite natural to try to stay as concrete as possible from the very beginning.

Many other interesting questions then emerge; for example, if correlation functions

1. In particular, Kyt¨ol¨a and Ridout have recently managed to classify rigorously a special class of Virasoro representations calledstaggered modules[59], following the pioneering work of Rohsiepe [49].

These modules seem to play a special role in the physics of boundary LCFTs, and it is for example quite satisfying to see that the 𝑏-number of Gurarie [50] can actually be computed from a purely algebraic viewpoint [57]. We shall come back to these staggered modules in Chapter 2.

such as (1.4) do appear in physical systems, then what do they describe precisely?

What kind of physical observables are logarithms related to? Can we understand indecomposability more physically? We will see that studying directly concrete lattice models provides reasonably-satisfying partial answers to these questions.

1.4 Organization of the manuscript

The outline of the remainder of this thesis is as follows:

∙ Chapter 2 contains an introduction to Logarithmic CFTs and indecomposability, assuming that the reader has a basic knowledge of CFTs only. All the rele- vant algebraic concepts (indecomposable representations, staggered modules for Virasoro etc.) are introduced using concrete examples. This chapter does not contain any new result per se – although it does reproduce some of the opera- tor product expansion arguments presented in [71, 72], but hopefully it provides a self-contained review of the field with examples worked out in details, that will take the reader from logarithmic correlations to complicated indecomposable Virasoro representations.

∙ Chapter 3 is a review of how lattice models can provide regularizations of Loga- rithmic CFTs, using the ideas of Read and Saleur [68, 69] – relying quite heavily on the recent review [73]. We explain how to measure indecomposability param- eters [71] and how to compute fusion rules [72] directly from the lattice.

∙ In chapter 4, we push further the ideas of the previous chapter to attempt a classification of Virasoro indecomposable representations relevant for physical applications [74]. We also study non-chiral (bulk) LCFTs, that are obtained as scaling limits of periodic lattice models, mostly from the point of view of inde- composability parameters [75]. Some (yet) unpublished results on the relation between open and periodic lattice models [76] and periodic percolation are also discussed.

∙ In the last chapter 5, we first discuss why LCFTs are relevant to describe crit- ical points in quenched disordered systems using supersymmetry or the replica trick [18]. We also adapt the ideas of Cardy [18, 77] to obtain geometrical ob- servables in the Potts model – and in particular in percolation – that behave logarithmically at the critical point. The 𝑂(𝑛) model, dilute polymers and span- ning trees are also discussed from the perspective of LCFTs. This chapter is based on the papers [78, 79].

Chapter 2

A short introduction to Logarithmic Conformal Field Theory

In this chapter, we will review the main ingredients of Logarithmic CFTs. Although conformal invariance yields several general results,e.g. the form of two and three-point correlators [80], in any dimension, most of the successes of CFTs are restricted to𝑑= 2 dimensions (see e.g.[61, 81]), where the conformal group becomes infinite dimensional.

We will henceforth restrict ourselves to two dimensions, mainly to fix some notations that will be used throughout this thesis. However, note that in principle, LCFTs exist in any dimension, and some of the results of this section can actually be readily generalized to higher dimensions. We shall come back to higher dimensional LCFTs in Chapter 5.

We will begin this chapter by coming back to CFT basics, and analyze how scale invariance can be made compatible with logarithmic correlations. We will show that logarithms are related to Jordan cells in the scale transformation generator, and we shall discuss the general form of the logarithmic correlators [45]. At this point, we will introduce the indecomposability parameters or 𝑏-numbers that characterize the logarithmic structure of a LCFT. The second section of this chapter is a review of the ‘𝑐 → 0 catastrophe’ [50, 54], and contains a discussion of a general pattern to explain how LCFTs can be thought of as limits of ordinary (non-logarithmic but ir- rational) CFTs¹. Within this approach, logarithms can be shown to arise as limits of power-law correlations, thus yielding simple formula for the indecomposability param- eters [71]. In order to relate these results to more formal algebra, we define and give simple examples of indecomposable representations in the case of the Lie superalgebra 𝔤𝔩(1∣1). The symplectic fermions theory [51, 52] will then provide a concrete exam- ple to illustrate all these new concepts: logarithms, indecomposable representations, indecomposability parameters, etc. Finally, the end of this chapter will be devoted to the ubiquitous Virasoro staggered modules – a class of indecomposable Virasoro modules especially important to LCFTs. This will also give us a chance to define inde- composability parameters in a purely algebraic language, as a number characterizing a Virasoro representation.

1. This idea will be used extensively in Chapter 5.

In this chapter, the emphasis will be on (L)CFTs rather than on lattice models. We will therefore try not to think of CFTs as describing properties of statistical models for a while, but rather consider them as abstract field theories that satisfy conformal invariance. We will go back to lattice models and show how all the results of this chapter are related to interesting physical problems in the next chapters.

2.1 Logarithmic operators, scale invariance and in- decomposability parameters

In this first section, we define Logarithmic operators and show that logarithmic cor- relations do not break conformal invariance. We derive the general form of two-point functions in the case of a rank-two Jordan cell in the scale transformation generator, and discuss the case of higher-rank Jordan cells. Furthermore, we introduce indecom- posability parameters, also called logarithmic couplings or 𝑏-numbers.

2.1.1 Conformal invariance in 2D, primary operators and Ward identity

We start with some general well-known results on ordinary (non logarithmic) CFTs.

More details can be found in [61] or in the seminal paper [4].

Conformal transformations

A conformal transformation is defined as an invertible mapping ⃗𝑥 7→⃗𝑥˜ which pre- serves the form of the metric tensor 𝑔_𝜇𝜈(⃗𝑥), up to a local scale Λ(⃗𝑥)

𝑔𝜇𝜈(⃗𝑥) = Λ(⃗𝑥)𝑔˜ 𝜇𝜈(⃗𝑥). (2.1) These transformations form the conformal group, which contains the Poincar´e group as a subgroup (Λ(⃗𝑥) = 1), as well as scale transformations (Λ(⃗𝑥) = Λ). Conformal invari- ance can be thought of as a local version of scale invariance. In generic dimension 𝑑, the conformal group can be shown to be isomorphic to 𝑆𝑂(𝑑+ 1,1), and the covariance of the correlation functions under this finite dimensional Lie group already enforces the form of two and three-point correlators [80]. However, in two dimensions², eq. (2.1) re- duces to the Cauchy-Riemann equations and any analytical mapping 𝑤(𝑧) ( ¯∂𝑤 = 0) is conformal, i.e. preserves angles in 2D (see Fig. 2.1). Strictly speaking, these conformal transformations do not have to be well-defined everywhere and invertible, which leads to the distinction between local and global conformal transformations. The subset of global invertible transformations form the group of global conformal transformations 𝑆𝐿(2,ℂ)/ℤ2 ≃𝑆𝑂(3,1).

2. We will use complex coordinates 𝑧 = 𝑥⁰+𝑖𝑥¹ and ¯𝑧 =𝑥⁰−𝑖𝑥¹, with ∂ = ¹₂(∂0−𝑖∂1) and

∂¯= ¹₂(∂0+𝑖∂1).

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

Figure 2.1: A rectangular grid (left) and its image under the conformal transformation 𝑓(𝑧) = ^𝑧₃² (right). An important point is that 𝑓(𝑧) preserves angles as can clearly be seen on these pictures.

Primary operators

CFTs are quantum field theories with conformal transformations as symmetry. The fundamental fields of a CFT are the so-called primary operators𝜙_ℎ,ℎ¯(𝑧,𝑧) which trans-¯ form covariantly under a conformal map 𝑧 7→𝑤(𝑧), ¯𝑧 7→𝑤(¯¯ 𝑧)

𝜙˜_ℎ,¯ℎ(𝑤,𝑤) =¯ (𝑑𝑤

𝑑𝑧

)₋ℎ( 𝑑𝑤¯

𝑑¯𝑧 )₋^¯ℎ

𝜙_ℎ,¯ℎ(𝑧,𝑧),¯ (2.2) where ℎ,¯ℎ are the conformal weights of the field 𝜙. In order to recover some more physically transparent quantities, we define

Δ = ℎ+ ¯ℎ, (2.3a)

𝑠 = ℎ−¯ℎ. (2.3b)

Δ is the usual scaling dimension that controls how 𝜓 transforms under a scale trans- formation⃗𝑥 7→Λ⃗𝑥

𝜙(Λ⃗𝑥) = Λ˜ ^−Δ𝜙(⃗𝑥), (2.4)

whereas𝑠is the conformal spin which appears naturally when rotating the system with an angle 𝜃

𝜙(˜ ℛ𝜃⃗𝑥) = e^𝑖𝜃𝑠𝜙(⃗𝑥). (2.5)

Note that if 𝜙 transforms as (2.2) (i.e. is primary), its derivatives will not be primary and will have more complicated transformation laws under conformal transformations.

Stress-energy tensor and Ward identity

A fundamental object in a CFT is the stress-energy tensor 𝑇^𝜇𝜈, defined through the variation of the action 𝑆 under an arbitrary transformation of the coordinates 𝑥^𝜇7→𝑥^𝜇+𝜖^𝜇

𝛿𝑆 = 1 2𝜋

∫

d²𝑥 𝑇^𝜇𝜈 ∂_𝜇𝜖_𝜈. (2.6)

The stress-energy tensor measures the reaction of the system with respect to a change in geometry. It can be chosen to be symmetric, and invariance with respect to rotations and translations implies, in complex coordinates, that ∂𝑇¯ = ¯∂𝑇 = 0 with 𝑇(𝑧) = 𝑇_𝑧𝑧 and ¯𝑇(¯𝑧) =𝑇𝑧¯¯𝑧. Scale invariance makes the stress-energy tensor traceless, so that𝑇𝑧¯𝑧 = 𝑇𝑧𝑧¯ = ¹₄𝑇_𝜇^𝜇 = 0. In an ordinary CFT, the holomorphic and antiholomorphic dependence decouple, and the whole theory can be constructed as sum of tensor products of chiral and anti-chiral sectors³. We will thus forget about the antiholomorphic sector for now, and focus on infinitesimal holomorphic conformal transformations𝑧 7→𝑤(𝑧) =𝑧+𝜖(𝑧).

Using (2.2), a primary field 𝜙ℎ with holomorphic conformal weight ℎ transforms as

𝛿𝜙_ℎ =𝜖∂𝜙_ℎ +ℎ𝜙_ℎ∂𝜖. (2.7)

The variation of correlation functions involving 𝜙ℎ under this infinitesimal conformal transformation can also be expressed in terms of a contour integral involving the stress- energy tensor 𝑇(𝑧)

𝛿𝜙ℎ =

∮

𝑧

d𝜉

2𝜋𝑖𝜖(𝜉)𝑇(𝜉)𝜙ℎ(𝑧), (2.8)

where this equality should be understood as holding when inserted in a correlator.

From (2.7) and (2.8), one obtains the following Operator Product Expansion(OPE) – called conformal Ward identity,

𝑇(𝑧)𝜙ℎ(𝑤)∼ ℎ

(𝑧−𝑤)²𝜙ℎ(𝑤) + 1

𝑧−𝑤∂𝜙ℎ(𝑤), (2.9)

where we dropped less singular contributions. Let us also define the Virasoro modes 𝐿𝑛 as the Laurent series

𝑇(𝑧) =∑

𝑛∈ℤ

𝐿_𝑛

𝑧^𝑛+2. (2.10)

3. This will not hold anymore for a LCFT, where the holomorphic and antiholomorphic sectors can be glued in a highly non-trivial fashion.

We also define the antiholomorphic modes ¯𝐿𝑛from ¯𝑇(¯𝑧) in the same way. The Virasoro modes 𝐿𝑛and ¯𝐿𝑛satisfy the commutation relations of the Virasoro algebra (1.2), with [𝐿_𝑛,𝐿¯_𝑚]

= 0. The Virasoro algebra structure is encoded in the following OPE 𝑇(𝑧)𝑇(𝑤)∼ 𝑐/2

(𝑧−𝑤)⁴ + 2

(𝑧−𝑤)²𝑇(𝑤) + 1

𝑧−𝑤∂𝑇(𝑤), (2.11) where we recall that 𝑐 is the central charge of the theory. The stress-energy tensor provides an example of quasiprimary field, which satisfies (2.2) only for global conformal transformations.

Within the usual radial quantization framework, the vacuum state∣0⟩is defined such that 𝐿_𝑛∣0⟩= 0 for all 𝑛≥ −1. This implies that the vacuum must be invariant under global conformal transformations, spanned by𝐿₋1,𝐿0 and𝐿1. The Ward identity (2.9) then implies that the asymptotic state ∣𝜙ℎ⟩ = lim𝑧→0𝜙ℎ(𝑧)∣0⟩ is an eigenstate of 𝐿0

with eigenvalue ℎ, and 𝐿𝑛∣𝜙ℎ⟩= 0 for 𝑛 >0. The operator 𝐿0 in the chiral theory, or rather 𝐿0 + ¯𝐿0 in the full theory, generates the scale transformations (or dilatations)

⃗𝑥7→Λ⃗𝑥, which are nothing but time translations in radial quantization. Hence,𝐿0 (or 𝐿0+ ¯𝐿0) plays the role of the Hamiltonian of the system.

2.1.2 Logarithmic operators and two-point functions

Let us now turn to the definition of logarithmic operators [45] (see also the recent review [82] and references therein), and compute their two-point correlation functions.

Logarithmic operators

For the sake of the argument, we shall focus on the holomorphic sector only. We will come back to non-chiral correlation functions later. Let𝜙 be a primary field with conformal weight ℎ. The associated state ∣𝜙⟩ is thus an eigenstate of the Hamiltonian 𝐿0– the scale transformations generator: 𝐿0∣𝜙⟩=ℎ∣𝜙⟩. It satisfies𝐿1∣𝜙⟩=𝐿2∣𝜙⟩= 0.

Let us now imagine that there exists a field 𝜓 which satisfies 𝑇(𝑧)𝜓(𝑤)∼ ⋅ ⋅ ⋅+ ℎ𝜓+𝜙

(𝑧−𝑤)² + 1

𝑧−𝑤∂𝜓, (2.12)

so that𝐿₀∣𝜓⟩ =ℎ∣𝜓⟩+∣𝜙⟩, and𝐿_𝑛>0∣𝜓⟩ ∕= 0 in general. We will nevertheless assume that 𝐿1∣𝜓⟩= 0. The field 𝜓(𝑧) is called logarithmic partner of 𝜙. Concrete examples of such fields will be given later. In the basis (∣𝜙⟩,∣𝜓⟩), Hamiltonian reads

𝐿0 =

(ℎ 1 0 ℎ

)

. (2.13)

It has a rank-two Jordan cell and is therefore non-diagonalizable. We will say that the two fields 𝜙 and 𝜓 are mixed by 𝐿₀ into a rank-two Jordan cell. Under global

infinitesimal conformal transformations 𝑤(𝑧) =𝑧+𝜖(𝑧)⁴, they transform as

𝛿𝜙 = 𝜖∂𝜙+ℎ𝜙∂𝜖, (2.14a)

𝛿𝜓 = 𝜖∂𝜓+ (ℎ𝜓+𝜙)∂𝜖. (2.14b)

Let us emphasize again that these equations hold if 𝜖(𝑧) = 𝑎+𝑏𝑧 +𝑐𝑧², that is, if 𝑤(𝑧) = 𝑧+𝜖(𝑧) is a global conformal transformation. In general, the transformation laws are more complicated and involve higher derivatives of 𝜖(𝑧); in particular, recall that 𝜓 does not transform as (2.14) for an arbitrary conformal transformation.

After a finite scale transformation 𝑧 7→Λ𝑧, one has ( 𝜓(Λ𝑧)˜

𝜙(Λ𝑧)˜ )

= Λ

−

⎛

⎝

ℎ 1 0 ℎ

⎞

⎠( 𝜓(𝑧) 𝜙(𝑧)

)

=

( Λ^−ℎ(𝜓(𝑧)−𝜙(𝑧) log Λ) Λ^−ℎ𝜙(𝑧)

)

. (2.15) Physically, this means that after a scale transformation, or after a Renormalization Group (RG) transformation, the field 𝜓(𝑧) will be mixed with the scaling field 𝜙(𝑧), and there is of course no way to change the field basis to get rid of this mixing. Non- diagonalizable RG flows are allowed because of the non-unitarity of the theory, and although it might seem quite exotic at first sight, we will see in the next chapters that such logarithmic fields are quite common is physics.

Note that because the Hamiltonian is non-diagonalizable, it is also non hermitian so we can see at this point that if such logarithmic partner fields do exist, the CFT cannot be unitary. Nevertheless, it is worth pointing out that one still has 𝐿^†₀ = 𝐿0

for the usual Virasoro bilinear form 𝐿^†_𝑛 = 𝐿₋𝑛, by definition, but the bilinear form † is no longer positive definite. For instance, we will show in the next paragraph that conformal invariance leads to ⟨𝜙∣𝜙⟩= 0.

Logarithmic two-point functions

Now that we know how these logarithmic fields transform under global conformal transformations, we would like to compute their two-point correlation functions – three- point functions could also be considered but we will restrict to two-point function for the sake of simplicity. Just like in ordinary CFT, global conformal invariance fixes the form of two and three-point correlators in any dimension [80], but we will restrict to two dimensions for now, postponing our discussion of higher dimensional LCFTs to Chap. 5.

First of all, because of translation invariance (𝜖 = 𝑎), the correlators ⟨𝜙(𝑧)𝜙(𝑤)⟩,

⟨𝜙(𝑧)𝜓(𝑤)⟩and⟨𝜓(𝑧)𝜓(𝑤)⟩depend only on𝑢=𝑧−𝑤. We will denote by𝑓(𝑢),𝑔(𝑢) and ℎ(𝑢) the corresponding functions, that is,⟨𝜙(𝑧)𝜙(𝑤)⟩=𝑓(𝑧−𝑤),⟨𝜙(𝑧)𝜓(𝑤)⟩=𝑔(𝑧−𝑤) and ⟨𝜓(𝑧)𝜓(𝑤)⟩ = ℎ(𝑧 −𝑤). Note that we already anticipated that ⟨𝜙(𝑧)𝜓(𝑤)⟩ =

⟨𝜓(𝑧)𝜙(𝑤)⟩ for the sake of simplicity. Scale invariance (𝜖 = 𝑎𝑧) then implies the

4. Translations correspond to the choice𝜖(𝑧) =𝑎, dilatations to 𝜖(𝑧) =𝑎𝑧, and special conformal transformations to𝜖(𝑧) =𝑎𝑧², with𝑎∈ℂ.

following differential equations

𝑢𝑑𝑓

𝑑𝑢 + 2ℎ𝑓 = 0, (2.16a)

𝑢𝑑𝑔

𝑑𝑢 + 2ℎ𝑔+𝑓 = 0, (2.16b)

𝑢𝑑𝑘

𝑑𝑢 + 2ℎ𝑘+ 2𝑔 = 0. (2.16c)

Similarly, it is straightforward to show that special conformal transformations yield equations that are compatible with (2.16) if and only if 𝑓(𝑢) = 0. The remaining equations can readily be solved, and one ends up with

⟨𝜙(𝑧)𝜙(0)⟩ = 0, (2.17a)

⟨𝜙(𝑧)𝜓(0)⟩ = 𝑏

𝑧^2ℎ, (2.17b)

⟨𝜓(𝑧)𝜓(0)⟩ = 𝜃−2𝑏log𝑧

𝑧^2ℎ , (2.17c)

where 𝜃 and 𝑏 are some constants. Let us assume for the sake of the argument that the normalization of𝜙 can be fixed in some way, then the normalization of 𝜓 is given⁵ by the equation 𝐿0𝜓 = ℎ𝜓 +𝜙. In that case, while the constant 𝜃 is arbitrary and can be canceled by a choice 𝜓 → 𝜓− _2𝑏^𝜃𝜙, the parameter 𝑏 is a fundamental number that characterizes the structure of the Jordan cell. It is unique and well-defined once a given normalization of the field 𝜙 has been chosen⁶. We also emphasize that the logarithmic term in the correlation function ⟨𝜓(𝑧)𝜓(0)⟩ is perfectly consistent with scale and conformal invariance, but it implies that 𝐿0 is non-diagonalizable because log Λ𝑧 = log Λ + log𝑧. It is also important to remark that 𝜙(𝑧) must be a null-field by conformal invariance, that is to say, introducing the usual Virasoro bilinear form,

⟨𝜙∣𝜙⟩ = 0⁷. However, 𝜙(𝑧) does not decouple as the correlator ⟨𝜙(𝑧)𝜓(0)⟩ does not vanish. This is crucially different from what we are used to with ordinary CFTs.

Finally, let us point out that logarithmic partners are unique: this can be argued using conformal invariance, or more simply by remarking that if 𝐿0∣𝜓1⟩ =ℎ∣𝜓1⟩+∣𝜙⟩ and 𝐿0∣𝜓2⟩ =ℎ∣𝜓2⟩+∣𝜙⟩, then the combination ∣𝜓1⟩ − ∣𝜓2⟩ decouples and there remains only one true logarithmic partner (∣𝜓₁⟩+∣𝜓₂⟩)/2. However, higher-rank Jordan cells are allowed (see next paragraph).

Note that similar equations can be obtained in the non-chiral case, but the impor- tant point is that 𝐿0 −𝐿¯0 has to remain diagonalizable so that the theory remains local [83]. Therefore, if 𝐿₀ is non-diagonalizable, then so is ¯𝐿₀. Two-point correlation functions are readily obtained by replacing log𝑧→log∣𝑧∣², and 𝑧^2ℎ →𝑧^2ℎ𝑧¯^2¯ℎ.

5. Note however that there still remains a degree of freedom𝜓→𝜓+𝛼𝜙in the definition of𝜓.

6. Anticipating a little bit, in most (but not all) cases of interest, 𝜙 is a descendant of another primary field. The normalization is then be given by the way 𝜙 is related to this ‘parent’ primary field, see eq. (2.21).

7. Note also that⟨𝜓∣𝜓⟩=∞.